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On licci squarefree monomial ideals

Om Prakash Bhardwaj, Trung Chau, Omkar Javadekar

Abstract

We study the licci property for several classes of squarefree monomial ideals arising from graphs and related combinatorial structures. We characterize licci bi-Cohen-Macaulay squarefree monomial ideals, complementary edge ideals, $t$-path ideals of cycles, and $(t-1)$-suspensions of graphs. Consequently, the full list of licci path ideals of trees is obtained. This work extends the known classification of licci edge ideals to a broader family of path ideals.

On licci squarefree monomial ideals

Abstract

We study the licci property for several classes of squarefree monomial ideals arising from graphs and related combinatorial structures. We characterize licci bi-Cohen-Macaulay squarefree monomial ideals, complementary edge ideals, -path ideals of cycles, and -suspensions of graphs. Consequently, the full list of licci path ideals of trees is obtained. This work extends the known classification of licci edge ideals to a broader family of path ideals.
Paper Structure (5 sections, 18 theorems, 59 equations)

This paper contains 5 sections, 18 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Licci bi-Cohen--Macaulay ideals
  4. Licci complementary edge ideals
  5. Licci path ideals of graphs

Key Result

Theorem 1.1

Let $I$ be a bi-Cohen--Macaulay squarefree monomial ideal in a polynomial ring $S$ over a field. Let $\mathfrak{m}$ denote the homongeneous maximal ideal of $S$. Then $I_{\mathfrak{m}}$ is licci in $S_{\mathfrak{m}}$ if and only if $\operatorname{ht}(I) \leq 2$ or $I$ is generated by variables.

Theorems & Definitions (47)

  • Theorem 1.1: \ref{['thm:licci-biCM']}
  • Theorem 1.2: \ref{['thm:licci-complementary']}
  • Theorem 1.3: \ref{['thm:licci-path-ideals-suspension']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 37 more